Numerical analysis of a semilinear fractional diffusion equation

TL;DR

This paper develops error estimates for numerical solutions of a semilinear fractional diffusion equation with nonsmooth data, introducing new inequalities and analyzing both spatial and temporal discretizations.

Contribution

It introduces a new Grönwall's inequality and its discrete form, deriving optimal error estimates for semilinear fractional diffusion equations with nonsmooth initial data.

Findings

Error estimates in Sobolev norms are optimal with respect to solution regularity.

A sharp temporal error estimate on graded grids is established.

Spatial accuracy in the L2 norm is achieved as O(h^2(t^{-eta} + log(1/h))).

Abstract

This paper considers the numerical analysis of a semilinear fractional diffusion equation with nonsmooth initial data. A new Gr\"onwall's inequality and its discrete version are proposed. By the two inequalities, error estimates in three Sobolev norms are derived for a spatial semi-discretization and a full discretization, which are optimal with respect to the regularity of the solution. A sharp temporal error estimate on graded temporal grids is also rigorously established. In addition, the spatial accuracy $\scriptstyle O(h^2(t^{-\alpha} + \ln(1/h)\!)\!) $ in the pointwise $ \scriptstyle L^2(\Omega) $-norm is obtained for a spatial semi-discretization. Finally, several numerical results are provided to verify the theoretical results.